Find the derivative of yx+5y-sin(y) = x

When differentiating an expression of x and y we can use implicit differentiation when terms contain the y variable, recall that this is essentially differentiating the y term by y and then multiplying by dy/dx as d(f(y))/dx = d(f(y))/dy * dy/dx = f'(y) * dy/dx. We start by start by differentiating both sides in terms of dx: d(yx+5y-sin(y))/dx = d(x)/dx The left hand side can be split into d(yx)/dx + d(5y)/dx - d(sin(y))/dx, the right hand side is 1 by power rule. (1x^1-1 = 1x⁰ = 1)We then deal with the left hand side term by term:1). For the first term we can use the product rule: d((f * g)(x))/dx = g * d(f(x))/dx + f * d(g(x))/dx which simplifies to x * dy/dx + y 2). For the second term we can directly use implicit differentiation d(5y)/dx = 5*dy/dx3). For the third term we can use implicit differentiation: d(sin(y))/dx = cos(y)dy/dx as derivative of sin is cos.4). Finally put it all together: y + (dy/dx)(x+5-cos(y)) = 1, rearrange by subtracting y and dividing by (x+5-cos(y)) and we finally get dy/dx = (1-y)/(x+5-cos(y))